Interquartile range
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In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically Hspread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles,^{[1]}^{[2]} IQR = Q_{3} − Q_{1}. In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is the most significant basic robust measure of scale.
The IQR is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rankordered data set into four equal parts. The values that separate parts are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.
Contents
Use
Unlike total range, the interquartile range has a breakdown point of 25%,^{[3]} and is thus often preferred to the total range.
The IQR is used to build box plots, simple graphical representations of a probability distribution.
For a symmetric distribution (where the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD).
The median is the corresponding measure of central tendency.
The IQR can be used to identify outliers (see below).
The quartile deviation or semiinterquartile range is defined as half the IQR.^{[4]}^{[5]}
Algorithm
Quartiles are calculated recursively, by using median.^{[6]}
If the number of entries is an even number 2n, then the first quartile Q_{1} is defined as
 first quartile Q_{1} = median of the n smallest entries
and the third quartile Q_{3} = median of the n largest entries^{[6]}
If the number of entries is an odd number 2n+1, then the first quartile Q_{1} is defined as
 first quartile Q_{1} = median of the n+1 smallest entries
and the third quartile Q_{3} = median of the n+1 largest entries^{[6]}
The second quartile Q_{2} is the same as the ordinary median.^{[6]}
Examples
Data set in a table
The following table has 13 rows, and follows the rules for the odd number of entries.
i  x[i]  Median  Quartile 

1  7  Q_{2}=87 (median of whole table) 
Q_{1}=31 (median of upper half, from row 1 to 7) 
2  7  
3  31  
4  31  
5  47  
6  75  
7  87  
Q_{3}=119 (median of lower half, from row 7 to 13) 

8  115  
9  116  
10  119  
11  119  
12  155  
13  177 
For the data in this table the interquartile range is IQR = Q_{3} − Q_{1} = 119  31 = 88.
Data set in a plaintext box plot
+++ o *    +++ +++++++++++++ number line 0 1 2 3 4 5 6 7 8 9 10 11 12
For the data set in this box plot:
 lower (first) quartile Q_{1} = 7
 median (second quartile) Q_{2} = 8.5
 upper (third) quartile Q_{3} = 9
 interquartile range, IQR = Q_{3}  Q_{1} = 2
 lower 1.5*IQR whisker = Q_{1}  1.5 * IQR = 7  3 = 4
 upper 1.5*IQR whisker = Q_{3} + 1.5 * IQR = 9 + 3 = 12
Distributions
The interquartile range of a continuous distribution can be calculated by integrating the probability density function (which yields the cumulative distribution function—any other means of calculating the CDF will also work). The lower quartile, Q_{1}, is a number such that integral of the PDF from ∞ to Q_{1} equals 0.25, while the upper quartile, Q_{3}, is such a number that the integral from ∞ to Q_{3} equals 0.75; in terms of the CDF, the quartiles can be defined as follows:
where CDF^{−1} is the quantile function.
The interquartile range and median of some common distributions are shown below
Distribution  Median  IQR 

Normal  μ  2 Φ^{−1}(0.75)σ ≈ 1.349σ ≈ (27/20)σ 
Laplace  μ  2b ln(2) ≈ 1.386b 
Cauchy  μ  2γ 
Interquartile range test for normality of distribution
The IQR, mean, and standard deviation of a population P can be used in a simple test of whether or not P is normally distributed, or Gaussian. If P is normally distributed, then the standard score of the first quartile, z_{1}, is 0.67, and the standard score of the third quartile, z_{3}, is +0.67. Given mean = X and standard deviation = σ for P, if P is normally distributed, the first quartile
and the third quartile
If the actual values of the first or third quartiles differ substantially^{[clarification needed]} from the calculated values, P is not normally distributed. However, a normal distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and 0.67 and not be normally distributed (so the above test would produce a false positive). A better test of normality, such as QQ plot would be indicated here.
Outliers
The interquartile range is often used to find outliers in data. Outliers here are defined as observations that fall below Q1 − 1.5 IQR or above Q3 + 1.5 IQR. In a boxplot, the highest and lowest occurring value within this limit are indicated by whiskers of the box (frequently with an additional bar at the end of the whisker) and any outliers as individual points.
See also
References
 ^ Upton, Graham; Cook, Ian (1996). Understanding Statistics. Oxford University Press. p. 55. ISBN 0199143919.
 ^ Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. ISBN 1584880597 page 18.
 ^ Rousseeuw, Peter J.; Croux, Christophe (1992). Y. Dodge, ed. "Explicit Scale Estimators with High Breakdown Point" (PDF). L1Statistical Analysis and Related Methods. Amsterdam: NorthHolland. pp. 77–92.
 ^ Yule, G. Udny (1911). An Introduction to the Theory of Statistics. Charles Griffin and Company. pp. 147–148.
 ^ Weisstein, Eric W. "Quartile Deviation". MathWorld.
 ^ ^{a} ^{b} ^{c} ^{d} Bertil., Westergren, (1988). Beta [beta] mathematics handbook : concepts, theorems, methods, algorithms, formulas, graphs, tables. Studentlitteratur. p. 348. ISBN 9144250517. OCLC 18454776.